Cluster formation due to repulsive spanning trees in attractively coupled networks

Ensembles of coupled nonlinear oscillators are a popular paradigm and an ideal benchmark for analyzing complex collective behaviors. The onset of cluster synchronization is found to be at the core of various technological and biological processes. The current literature has investigated cluster synchronization by focusing mostly on the case of attractive coupling among the oscillators. However, the case of two coexisting competing interactions is of practical interest due to their relevance in diverse natural settings, including neuronal networks consisting of excitatory and inhibitory neurons, the coevolving social model with voters of opposite opinions, and ecological plant communities with both facilitation and competition, to name a few. In the present article, we investigate the impact of repulsive spanning trees on cluster formation within a connected network of attractively coupled limit-cycle oscillators. We successfully predict which nodes belong to each cluster and the emergent frustration of the connected networks independent of the particular local dynamics at the network nodes. We also determine local asymptotic stability of the cluster states using an approach based on the formulation of a master stability function. We additionally validate the emergence of solitary states and antisynchronization for some specific choices of spanning trees and networks.

Figure 1

The complete graph $K_3$. We consider here the most straightforward nonbipartite connected network with $N=3$ nodes. The red dashed arcs represent one of the available spanning trees of $K_3$. This spanning tree is isomorphic to the other two available spanning trees.

Figure 2

Emergence of cluster synchronization in $K_3$. The introduction of the repulsive path (red dashed links) in $K_3$ as shown in Fig. 1 induces cluster synchrony. The system splits into two distinct groups. The vertex decomposition $V_1=\{1,3\}$ and $V_2=\left\{2\right\}$ of the repulsive spanning tree remains consistent with the cluster synchronization, where oscillators 1 and 3 belong to a group and the remaining oscillator 2 lies on the other cluster. The phase difference between these two clusters is $\pi$. Panel (a) shows the phase portrait, and panels (b)–(d) reveal the temporal evolution of $y_i, r_i$, and $x_i$, respectively. [(e)–(h)] The second row displays the snapshots at a specific iteration after the transients. For further information, please see the main text.

Figure 3

$K_4$ and its nonisomorphic spanning trees. The complete graph $K_4$ with four vertices possesses 16 spanning trees, from which the number of nonisomorphic spanning trees is 2. We draw $K_4$ in (a), and the other two in (b) and (c) represent those two nonisomorphic spanning trees. This figure is drawn using the software Gephi [62].

Figure 4

Predicting cluster synchronization in undirected and connected network $K_4$. (a) We pass the repulsive coupling through the spanning tree (red dashed arcs) shown in Fig. 3. The vertex decomposition of this spanning tree allows us to predict which oscillators belong to which clusters. The black arcs depict the links through which we pass the positive coupling strength ${\epsilon }_A=0.01$. Panel (b) assures the phase difference between these two clusters is exactly $\pi$. Panel (c) further confirms oscillator 1 belongs to one cluster, and the other three oscillators lie on another group. The temporal evolution of $y_i$ in (d) reveals the trajectories are oscillating in the limit-cycle regime maintaining cluster synchronization with $\pi$ phase difference.

Figure 5

Identifying the members of the clusters in $K_4$. (a) We choose a spanning tree (red dashed arcs) as shown in Fig. 3 of $K_4$. We pass the repulsive coupling with ${\epsilon }_R=-0.1$ through this spanning tree. (b) Oscillators 1 and 3 lie in one cluster, and the other two oscillators belong to the other cluster. Moreover, both of these clusters maintain a phase difference of $\pi$. (c) The vertex partitioning of the repulsive spanning tree predicts the members of the clusters without investigating the local dynamics. The snapshot of $x_i$ at a particular time after the transient validates our claim here. (d) The oscillating temporal evolution of the trajectories contemplates a phase difference of $\pi$ between these two clusters.

Figure 6

Numerically calculated frustration through two different repulsive spanning trees in $K_4$. We change the value of repulsive coupling strength ${\epsilon }_R$ in the range $[-0.03,0.01]$, starting from ${10}^{-2}$ with fixed step length $=-{10}^{-5}$. The initial conditions are drawn for each ${\epsilon }_R$ randomly within $[-1,1]\times [-1,1]$. The red (dark gray) line represents the result when the repulsive path is chosen as drawn in Fig. 3. The yellow (light gray) line portrays the impact of the spanning tree as shown in Fig. 3. Here the attractive coupling strength is kept fixed at ${\epsilon }_A=0.01$. $F$ deals with the frustration of the whole network, while the measure $F_{\mathrm{chain}}$ contains the information of the frustration only through the chain. The spanning tree in Fig. 3 is more effective than the spanning tree in Fig. 3 in terms of attaining the minimum value of $F$ and $F_{\mathrm{chain}}$. Every data point in this figure represents a single realization of the process.

Figure 7

Stability of cluster synchronization state. Cluster synchronization error $E$ (blue line) and the maximum Lyapunov exponent transverse to the cluster synchronization manifold $\mathrm{\Lambda }$ (red dashed line) as a function of repulsive coupling strength ${\epsilon }_R$. The corresponding network structure is illustrated in Fig. 5. The attractive coupling strength ${\epsilon }_A$ is kept fixed at 0.01. For further information, please see the main text.

Figure 8

Splitting of two distinct clusters in $K_{100}$. We choose the complete graph $K_{100}$ with $N=100$ vertices and 4950 edges and pass the repulsive coupling strength ${\epsilon }_R=-4.0$ through the simple chain containing only $N-1=99$ links. The positive coupling strength is kept fixed at ${\epsilon }_A=0.01$ and the initial conditions are randomly chosen from $[-1,1]\times [-1,1]$. The phase portrait in (a) and the temporal evolutions in (b) and (c) reveal the emergence of two clusters. The variation of the amplitude $r_i$ of each oscillator is manifested in (d). [(e)–(h)] The lower panel exhibits the snapshots at a particular time after the transients. These snapshots unveil the system settles down a two-cluster state, and the phase difference between these two clusters is $\pi$. Since the amplitude $r_i$ of all oscillators is equal, we observe the onset of antisynchronization.

Figure 9

The impact of a different spanning tree on $K_{100}$. (a) The phase portrait in the $xy$ plane, (b) the temporal evolution of $x_i$, and (c) the temporal evolution of $y_i$ ensure the system splits into two synchronized populations. (d) The temporal evolution of $r_i$ is represented in (d). [(e)–(h)] Due to our chosen spanning tree, oscillator 1 leaves the other synchronized population, and the phase of this oscillator deviates a distance of $\pi$ from the phase of the other oscillators. The snapshots in the (e)–(h) validate our claim of predicting the members of the clusters using the vertex partitioning of the used repulsive spanning tree. Here ${\epsilon }_R=-3.6$ and ${\epsilon }_A=0.01$.

Figure 10

The (local) stability of the cluster-synchronized state. The simulated maximum Lyapunov exponent $\mathrm{\Lambda }$ (red dashed line) transverse to the cluster synchronization manifold becomes negative, confirming each cluster's stability. The underlying network $K_{100}$ contains the chain as the repulsive subgraph as illustrated in Fig. 8. The stability of the synchronous solution is further validated by plotting the cluster synchronization error $E$ (blue line), which drops to zero beyond a specific strength of ${\epsilon }_R<0$. Here ${\epsilon }_A=0.01$. Please see the main text for further information.

Figure 11

Predicting the members of two synchronized clusters. (a) Red dashed arcs delineate a spanning tree of the drawn network with 10 nodes. We pass the positive coupling strength through the black edges. Here ${\epsilon }_A={10}^{-2}$ and ${\epsilon }_R=-{10}^{-1}$. (b) Utilizing the bipartiteness of the used repulsive spanning tree, we can easily predict the oscillators with indices $\{1,3,5,7,9\}$ and $\{2,4,6,8,10\}$ belong to two different clusters. In addition, the phase difference between any two representative oscillators of these respective two clusters is $\pi$. (c) The snapshot at a specific time after the transient validates the occurrence of cluster synchronization. (d) The temporal evolution of $y_i$ attests that the trajectories oscillate in a limit-cycle region, maintaining a fixed $\pi$ phase difference between these two synchronized populations.

Figure 12

Prediction of the members of the clusters through a different spanning tree. (a) We choose the same network as portrayed in Fig. 11 and pass the negative coupling strength ${\epsilon }_R=-{10}^{-1}$ through a different spaning tree contemplated through the red dashed arcs. The attractive coupling strength ${\epsilon }_A={10}^{-2}$ is applied through the black links. The vertex partitioning of the chosen repulsive path provides two disjoint and independent sets $V_1=\{1,4,5,8\}$ and $V_2=\{2,3,6,7,9,10\}$, respectively. Panels (b) and (c) confirm the members of the two clusters separate as per the vertex partitioning of the repulsive spanning tree. (d) The trajectories in the limit-cycle regime sustain coherent behavior within the same cluster; however, the clusters undergo a phase difference of $\pi$.

Figure 13

Numerically calculated frustration indices $F$ and $F_{\mathrm{chain}}$ for two nonisomorphic spanning trees. (a) Despite the chosen two spanning trees being different as depicted in Figs. 11 and 12, the emergent frustration in the whole network remains the same in the saturated domain. (b) However, the asymptotically saturated frustration that arises in the subgraph labeled as the chain in the article is zero (red) for the spanning tree shown in Fig. 11 and 0.889206886 (yellow) for the spanning tree used in Fig. 12. We vary the negative coupling strength ${\epsilon }_R$ from ${10}^{-2}$ with a tiny step size of $-{10}^{-5}$ in both these subfigures by keeping fixed the attractive coupling strength at ${\epsilon }_A=0.01$. Before the saturation, we observe random fluctuations in the values of these two measures $F$ and $F_{\mathrm{chain}}$ due to the choice of random initial conditions from $[-1,1]\times [-1,1]$ at each step. Every data point in this figure represents a single realization of the process.

Figure 14

A randomly chosen graph along with its two different spanning trees. (a) A nonbipartite network with 8 nodes and 15 links is drawn here. [(b) and (c)] We trace this network's two distinct nonisomorphic spanning trees as shown in (b) and (c). The vertex partitioning of the spanning tree in (b) is $V_1=\{1,3,5,7\}$ and $V_2=\{2,4,6,8\}$, whereas the set of vertices for the spanning tree in (c) is $V_1=\{1,5,6,7,8\}$ and $V_2=\{2,3,4\}$.

Figure 15

$F$ and $F_{\mathrm{chain}}$ with varying ${\epsilon }_R$. Numerically saturated values of $F$ and $F_{\mathrm{chain}}$ closely match with our theoretically calculated values of these measures. The red (dark gray) line represents the results for the repulsive spanning tree in Fig. 14, whereas the yellow (light gray) line displays the results for the spanning tree in Fig. 14. (a) For both of these nonisomorphic spanning trees, the system saturates to the $F\approx 0.666431904$, while our analytical predicted $F=2/3$. (b) Passing the repulsive coupling strength through the spanning tree in Fig. 14, we get $F_{\mathrm{chain}}\approx 0$. $F_{\mathrm{chain}}\approx 1.42900455$ for the spanning tree in Fig. 14, which is very close to the theoretically predicted $F_{\mathrm{chain}}=10/7$. Here ${\epsilon }_A={10}^{-2}$ and ${\epsilon }_R$ is varied with a small step length $-{10}^{-5}$ starting from ${10}^{-2}$. The initial conditions are drawn randomly from $[-1,1]\times [-1,1]$. Every data point in this figure represents a single realization of the process.

Figure 16

Cluster synchronization in a network of $N=6$ nonbipartite network. (a) We choose a nonbipartite network that does not contain the chain. We trace a spanning tree whose bipartiteness provides two disjoint sets $V_1=\{1,3,5\}$ and $V_2=\{2,4,6\}$. We pass the repulsive coupling strength ${\epsilon }_R=-0.1$ through the red dashed arcs and the attractive coupling strength ${\epsilon }_A=0.01$ through the black links. (b) If we choose one oscillator from the set $V_1$ and another one from the set $V_2$, then the phase difference between that two oscillators is $\pi$ as described through this snapshot. (c) The system splits into two groups, as revealed through the evolution of the oscillators in the $xy$ plane.

Figure 17

Predicting the members of the clusters through the vertex partitioning of the repulsive spanning tree. (a) The set of vertices of the chosen spanning tree (red dashed arcs) can be split into two disjoint sets $V_1=\{1,4,5\}$ and $V_2=\{2,3,6\}$. Here ${\epsilon }_A=0.01$ is passed through the black links of the chosen network of $N=6$ vertices. (b) The vertex decomposition of the repulsive bipartite subgraph allows predicting the two clusters appropriately. The snapshot verifies the $\pi$ phase difference between those two clusters. (c) The formation of two clusters separated by a phase difference of $\pi$ is portrayed here.

Figure 18

Variation of $F$ and Order with respect to ${\epsilon }_R$. We systematically vary the coupling strength ${\epsilon }_R$ with a fixed step length of 0.00001. For the forward transition, ${\epsilon }_R$ ranges from 0.01 to $-0.03$; conversely, for the backward transition, it ranges from $-0.03$ to 0.01. Initially, all conditions are randomly chosen from $[-1,1]\times [-1,1]$. Subsequently, the last point from each simulation iteration serves as the initial conditions for the next iteration, with a small perturbation added to each oscillator from the interval [0,0.01]. Notably, the coupling strength ${\epsilon }_R$ marking the forward transition ($F=2$ to $F<2$) differs from that of the backward transition ($F=1$ to $F>1$). This distinction is evident in (b), depicting the variation of the Kuramoto order parameter, denoted as Order. Despite multiple repetitions with different initial conditions, the observed transition remains consistent. Every data point in this figure represents a single realization of the process.